The present invention concerns position-measurement systems and methods, particularly methods of using range measurements to estimate position.
Some position measurement systems use radio signals broadcast from transmitters to measure the position of a radio receiver that receives the radio signals. A seminal example of one such system is a global positioning system.
The typical global positioning system (GPS) measures the three-dimensional, global position of a radio receiver. The receiver, sometimes mounted to a vehicle such as an automobile or aircraft, receives signals from a set of earth-orbiting satellite transmitters. Each signal indicates not only the position of its transmitter but also its is transmission time, enabling the receiver to approximate signal transit times and to estimate or measure the distances to the transmitters. A processor coupled to the receiver uses at least four of these measured distances, known as range measurements, to approximate or estimate the position of the receiver and the associated vehicle.
Estimating the position of the receiver from the range measurements generally entails solving a set of nonlinear range equations using an iterative calculation process. The iterative process entails starting with an initial position estimate, computing a second estimate using the initial estimate, computing a third estimate using the second estimate, and so on, with each successive estimate being better than the previous one and thus converging toward the actual receiver position. This process continues until the change between successive position estimates become insignificantly small.
Unfortunately, this iterative process is not only time consuming, but often fails to converge toward the actual receiver position when the receiver is not between the earth and the earth-orbiting satellite transmitters. Thus, for example, vehicles traveling in space outside the constellation of satellite transmitters cannot rely on a global positioning system for navigation.
In trying to solve these problems, others have sought to develop non-iterative, or closed-form, solutions to the range equations. See, for example, Bancroft and Chaffee, xe2x80x9cAn Algebraic Solution of the GPS Equation,xe2x80x9d IEEE Trans on Aerospace and Elect Systems, January 1985. However, this and other approaches have not been entirely acceptable, because they either fail to consistently converge on a single solution or give multiple solutions to the pseudorange equations, and thus require further analysis to pick the right one.
Accordingly, there remains a need for better ways of estimating position from range measurements.
To address this and other problems, the inventors devised new methods as well as systems and software embodying these methods, for using range measurements to estimate position. One exemplary method entails receiving a set of two or more range measurements; defining an error function based on the set of range measurements, with the error function having only one local minimum; and then determining a position estimate based the one local minimum of the error function. Other embodiments use this position estimate as an initial position estimate in an iterative process, such as Kalman filtering, to promote its rapid and consistent convergence to an appropriate position solution.
One exemplary application for this and other embodiments of the invention is GPS-based position estimation for spacecraft outside a GPS satellite constellation. Other applications include position estimation of cellular phones outside a triangle of three base towers, and in distributed robotic systems, position estimation of scout robots outside a triangle of three ranger robots.